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      Quantum ergodic restriction for Cauchy data: Interior QUE and restricted QUE

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          Abstract

          We prove a quantum ergodic restriction theorem for the Cauchy data of a sequence of quantum ergodic eigenfunctions on a hypersurface \(H\) of a Riemannian manifold \((M, g)\). The technique of proof is to use a Rellich type identity to relate quantum ergodicity of Cauchy data on \(H\) to quantum ergodicity of eigenfunctions on the global manifold \(M\). This has the interesting consequence that if the eigenfunctions are quantum unique ergodic on the global manifold \(M\), then the Cauchy data is automatically quantum unique ergodic on \(H\) with respect to operators whose symbols vanish to order one on the glancing set of unit tangential directions to \(H\).

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          surfaces

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            Quantum ergodicity for restrictions to hypersurfaces

            Quantum ergodicity theorem states that for quantum systems with ergodic classical flows, eigenstates are, in average, uniformly distributed on energy surfaces. We show that if N is a hypersurface in the position space satisfying a simple dynamical condition, the restrictions of eigenstates to N are also quantum ergodic.
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              Quantum Ergodic Restriction Theorems: Manifolds Without Boundary

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                Author and article information

                Journal
                01 May 2012
                2013-06-04
                Article
                1205.0286
                e780c42a-ba6c-41aa-87e9-c0d7241af356

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                History
                Custom metadata
                Math.Res.Lett 20 (2013) 465-475
                9 pages. Final version; incorporates referees' comments. To appear in MRL
                math.AP math-ph math.MP

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